Apparatus for effecting the transformations of spherical coordinates



July 28, 1925. 1,547,940

c. R. MONNEY APPARATUS FOR 'EFFEC'fING THE TRANSFORMATIONS OF SPHERICAL COORDINATES Filed Feb. 1924 5 Sheets-Sheet 1 Imeriar C .1? ,LQomne, \EyM s July 28, 1925. 1,547,940

C. R. MONNEY APPARATUS FOR EFFECTING THE TRANSFORMATIONS OF SPHERICAL COORDINATES Filed Feb. 26, 1924 3 Sheets-Sheet 5 3 1? 22 J It 1 z 1 1 11 jg Y A y 1 9 1 j; l I 05 v 1c 5 I 1C if I.

d 1 1 I E L} 1' J1 n" {a 1 Izwembr C; R moi-L Fatented July 28, 1925.

rsnaa CHARLES noennamoimny, orrAnIs, FRANCE.

APPARATUS FOR- EFFEGTING THE TRANSFORMATIONS OE SPI-IERICAIQ- COORDINATES.

Application filed February 26', 192.4. Serial'No. 695,363.

coordinates, more particularly applicable to I the execution of calculus of spherical trigonometry relating to the diurnalmovement of the stars and. in which the parallactic angle does notintervene.

This apparatus consists of the combination of two systems each similarly composed of two concentric circles the planes of which are at right angles; one of these circles rotates relatively to the other about a diameter at right angles to the plane of the first circle and carries a rider; both systems are concentrically arranged, so that the axes of rotationof the movable circles are in one and the same fixed plane, both systemsbeing capable of rotating relatively to each other about a diameter at right angles to this plane, so as to ensure their relative orientation, the riders being connectedby a joint the geometrical axis of which passes through the common centre of the circles.

Theapparatus forming the subject matter of the invention permit-s of the mechanical execution of all the problems which lead back to the following:

Five quantities being considered:

(Ir-The two coordinates of a point in one of the systems,

7).T'he two coordinates of this point in the other system;

a The relative orientation of the two systems.

To determine two of these quantities, the three others being known.

The applications of the apparatus are obviously very numerous. Byway ofexajmple, the following applications can be'cited:

o..Calculation ofthe position of a ship at sea (positionand orientation of the. line of altitude) 3,

b.Calculation of the azimuth bythe hour; V p I ci-Calculation of the azimuth by the altitude; V

vcZ.-l)etermination of a star knowing its altitude, its azimuth and the latitude of-the spot of observation;

e;Calculat-ion of the altitude of a star at a'given hour ina given spot;

- fi-Galculation of the hour of'rising and settingof-a star ata given spot.

The accompanying drawing illustrates, by way ofexample only a forn'iof carrying out the invention, the operation: of which will be'described hereafter in the particular case of calculation of a ships place at sea.

Fig. 1 is ail-explanatory diagram.

Fig.2. is asectional elevation'according to line A.'A of Fig. 4.

Fig.3 is a sectional'elevation according to line BB of Fig. 4. v

Fig. 4 is a plan View,

Fig. 5 is. a partialsectiomon an enlarged scale, according to line G() of Fig. 2,

Fig. 6 is. a partial section, on an enlarged scale, according to line D-D of Fig.2,

Fig. 7 is a partial section, on an enlarged scale, according to line EE of Fig. 2.

Fig. 8 is an elevation of the base or socle.

Fig. 9 is a side view, on anenlarged scale, of a detail. v

Referring to Fig. 1, O designates the present p osition,.of forv instance the ship on whichthe apparatusis placed, Z Z the vertical at O, N S the parallel at O at the axis of the world- The circle N Z S Z which is in the plane. of the drawing, is the meridian; circle, and N 'S E represents the horizon circle.v The circle El VV P" constitutes the circle of the equator, N A S the circle of declination.

In these conditions, the star A is situated in zenithalcoordinates by the zenithal distance Z A, .or. the altitude Aa which is the complement. thereof, and by the azimuth represented byrthe angle A Z P.

The local coordlnates of this star arethe polar distance N A, or the declination Ac which isthe complement thereof, .andthe horary angle N Z. I r On'theother hand, the relative orientation of the two systems of coordinates is given by the latitude, represented by the angle N O N. 7

Ifan apparatus indicating the-various circles illustrated is available, it will be possible, by knowing three of the previously indicated quantities, that is to say; the coordinates of a star in each of the systems and the relative orientation of these systems, to indicate this star and, consequently, to determine the position by simply reading the two other quantities.

The apparatus illustrated in Figs. 2 to 9 permits the accomplishment of these cond1 tions, but in this special form of construction, the whole of the device is simplified owing to the fact that the circles are replaced by half-circles. This simplificatlon offers no inconvenience in all the cases in which the angles are smaller than 7:, or when they are greater than 7:, but intervene, in the calculations usually executed, by their cosine. They can in fact be replaced, in this case, by the supplementary angle.

The apparatus rests on a plain circular base 1 strengthened on its upper face by ribs 1 and recessed at 1 for the sake of lightness. Columns or posts 1 are cast with the base 1 and support a ring, preferably of square cross section, and the upper face 1 of which is carefully ground, so as to form a plane. Two of the diametrically opposed columns, designated by 1 and l are wider than the others and are cut away at 1 and 1 for receiving bearings 2 and 2 which are secured by means of screws 3. The base 1 is provided. with a central boss or hub 1 on the top of which is arranged a pivot 1 of frusto-conical shape. The upper surface 1 of the boss or hub 1 is also ground so as to be perfectly plane and parallel to the surface 1 of the ring 1. The geometrical axis of the pivot 1 is exactly at right angles to the surface 1 and passes through the centre of the annular surface 1 A toothed segment 4 is secured on the columns 1, 1 and 1 by means of screws 5; the toothed segment 1- covers a half circumference and extends between the columns 1 and 1 On the other hand, a member 6, which is described more in detail hereafter, is mounted on the pivot 1. so as to be capable of pivoting on the latter. The lower face 6 ofthis member 6 is perfectly smooth and, when the member 6 is mounted on the pivot 1", the face 6 must come in contact with the surface 1 of the boss or hub 1". The member 6 has a circular plate 6 which extends over a little more than a half circumference, as clearly shown in Fig. 3. The faces 6 and 6 of this plate are parallel planes and the plane of the face 6 must pass, when the member- 6 is mounted on the pivot, through the geometrical axis of this pivot. On the other hand, the member. 6

has two bosses 6 and 6 which exteriorly are preferably of frustum shape and are perforated with cylindrical holes 6 and 6 The geometrical axes of the holes 6 and 6 are in a straight line, which latter meets the geometrical axis of the pivot 1 and is at right angles to the plane of the faces (3 and 6 This straight line thus constitutes the geometrical axis of the cylindrical sur.- face 6 limiting the circular plate 6*. Finally, long mortices 6 and 6 are provided in the bosses 6 and 6 these mortices opening in the cylindrical bores S and 6 The boss 6 has, at its lower part, plane lugs 6 and 6 the lower surface of which are plane; an arm 7 is provided with a plate 7 (Fig. 5) the upper surface of which corresponds to the lower surface of the lugs 6 and 6; the plate 7 is secured against these lugs and under the latter, for instance, by means of screws 7 The arm 7 is radially extended so as to form at its end a nose 7* the lower face 7 of which is plane and fits against the upper surface 1 of the ring 1 The upper face 7 of the nose 7 is inclined and carries a reference line 7; this line, if it was extended, would meet the geometrical axis of the pivot 1*.

From the foregoing description, it will be clearly seen that the member 6 can r0- tate on the pivot 1', the nose 7* moving along the upper surface 1 of the ring 1. For effecting this displacement with the necessary precision, the following means are preferably used:

A worm wheel 8 is mounted on a shaft 9 so as to be capable of rotating on the same; i this shaft is screw-threaded at 9 and is screwed in a threaded hole 7 of the arm 7. This arm has a U shaped cross section, as shown in Fig. 5. A worm 10, so cut as to gear with the worm wheel 8, is mounted on a shaft 11 on which it is secured by screws 10 and 10 the said shaft 11 passing through holes made in the lugs 7 and 7 of the arm 7 and carrying an operating knob 11. The worm wheel 8 also, gears with the toothed segment 4; so that by causing the worm 10 to rotate by means of the operating knob 11"-, the wheel 8 is caused to rotate and run' on the toothed segment 4, 4

thus carrying round the member 6 about the pivot 1 Cylindrical shafts 12 and 13 t in the bores 6 and G each of these shafts is perforated with a threaded hole in which are screwed threaded fingers 14 and 15 which pass through the mortices 6 and 6. The shafts 12 and 13 have a length slightly less than or at the most equal to the depth of the holes 6 and 6 and the threaded holes in which are screwed the threaded fingers 14 and 15 are so arranged that, when these fingers are brought back in the direction of the pivot l until they abut against the bottom of the corresponding mortices, the shafts 12 and 13 no longer project outside the bores 6 and 6 Screws 16 and 17, which are screwed in the threaded holes of thebosses 6 and 6 ensure of the arranging of the shafts 12 and 13 in any desired position.

A member 18, which is circular and the periphery of which covers a little more than a half circle carries two diametrally opposed bosses 18 an'd' 18" perforated with cylindrical holes 18 and 18 the geometrical axes of whichcoincide and form" a diameter; The inner faces 18 'and' 18 of the bosses 18" and 18 are plane; parallel and at right angles to the above mentioned diameter. They fit exactly against the faces 6"aiid 6 of the bossestl and 6. The shafts 12 and 13 enter the bores 18 and 18, so that, in theposition illustrated in Fig. 2, themei'n ber '18 may rotate on the said shafts 12 and 13. For effecting, this rotation with precision; the member 6 is provided with helical teeth formed on a circular projection 6*; a worm 19 gears with these teeth and it is mounted on a shaft 20 so as to rotate with this shaft. The shaft 20 carries for instance, forthat purpose; a journal 2O having a square cross section and on which is mounted the worm 19; The shaft 20 is j'ournalledih holes-provided in a support 21 which is secured on the member 18' by screws 22 and 23 for instance; the shaft 20 is moreover provided with an operating knob 24 which, for instance, is screwed on the screw threaded portion 20 of the said shaft. 7 v

The member 18 also carries an index 25 which issecured thereto-by means of screws 2'5 and 25 for instance. This index has two arms 25 and 25 connected by aflat part 25 9'in particular); the face12'5 of this flat part is, when the inde'x25 is mounted on the member 18, parallel to the face 6 of the plate 6 and arranged very near this face 6. The part 25 is bevelled at 25' and carries a reference line 25 which, if'it was extended, would meet the geometrical axis of the holes 6 and 6 On the member 18 is mounted a rider 26 which embraces the outer edges of the said member 18 through the two ledges 26'" and 26. The ledge 26 is bevelled. On the other hand, the rider 26 is recessed at 26 (Fig. 7; for receiving a worm 27 rigidly secured on the shaft 28 which on either side extends beyond the rider 26', so as to receive on either side an operating knob 28, 28 The worm 27 gears with helical teeth formed on a circular projection- 18 of the member 18. On the other hand, the rider 26 is provided with a cylindrical; hole 26 the geometrical axis of which, if it was extended, would constitute a radius of the member 18, the face 18 of which is smooth, the plane of this face passing through the com mon geometrical axis of the bores 66 In the cylindrical hole 26 fits a shaft 29 of another rider 29, which embraces the inner edges of a circular member 30, described hereafter. The rider 29 can therefore freely rotate relatively to the rider 26.

This rider 29 is bevelled at 29*, this oblique surface bearing a reference line 29.

The circular member 30, on the inner periphery of which the rider 29 moves, extends over a little more than a half circle and carries two trunnions 30 and 30 the geometrical axes of which are in alignment and constitute a diameter. These trunnions 30 and 30 are mounted in the bearings'2 and 2 and the above mentioned diameter is then in the same plane as the upper face 1 of the ring 1. On the other hand, the face 30 of the member 30 is smooth; its plane contains the above mentioned diameter which also meets the geometrical axis of the pivot 1 Finally, a circular member 31 is secured by means of the screws 32 and 33, on the ring 1 so that its middle plane is at right angles to the geometrical axis of the trunnions 30'.30 and contains the geometricalaxis of the pivot 1.

In the above describeddevice; the system of zenithal coordinates is indicated by the half-circle 31, which represents the horizon circle, and by the half-circle 30, which rep resents the circle of altitude; the geometrical axis of the trunnions -3'O-3O represents the vertical of thepresent position, for instance of the ship. The system of local coordinates is constituted by the half-circle 6", which represents the equatorial circle, and by the half-circle 18, which represents th'ecircle of declination. The star observed isindicatedby the whole of thetwo rider's 26 and 29. The geometrical-axis of the bores 6 and 6 represents the parallel to the axis of the world, traced through the spot where one happens to be. The surface Ie' represents the meridian circle.

The meridian circle la is graduated in degrees, but only on the half-circumference where the arm 7 moves. Besides the refer ence line 7,the surface 7 bearsthe scale or graduations of a Vernier permitting the reading of the half-minute. The half-circle 6 isgraduate'd in hours and the surface 25 bears the scale or graduations'of a Vernier ensuring of the reading of the second; The half-circle 18 is graduated in degrees and the surface 26 of the rider 26 carries a Vernier ensuring of the reading of the minute. Finally, the half-circle 30 is graduated in degrees.

In explaining the operation of the apparatus, the calculation of the position of a ship at sea will be'taken as example.

Itisknown that for finding the position of aship at sea, it is suflicient to draw a line of altitude. For that purpose, the altitude h. of a chosen star is observed at a given moment, the reckoned altitude he is then calculated; the difference between the two altitudes is searched and traced at the scale of the chart according to the horizontal direction where the star is perceived. This horizontal direction forms with the meridian an angle which is the azimuth of the star, which is observed or calculated. The line of altitude is the perpendicular drawn through the point found relatively to the horizontal direction traced.

The problem consists therefore in finding the altitude and azimuth of the star at the reckoned point. Now, the latitude cp of this point, the horary angle AI and the dec lination o: of the star at the moment are obviously known. These three indications render the star tangible; in fact, it suffices:

l.To place the reference mark 7 in latitude relatively to the geometrical'axis of the trunnions 30, the nose 7 moving along the meridian circle Ie when the knob 11 is operated.

2.--To place the half-circle of declination 18 in horary angle on the equatorial halfcircle 6*, by acting upon the knob 20 3.To place the rider 526 in declination on the circle of declination 18, by acting upon one of the two knobs 28 28'.

At this moment, the position of the rider 529 permits of the reading on the circle of altitude 30 the reckoned altitude and the position of this circle of altitude 3O relatively to the horizon circle 31 permits of the reading of the azimuth.

What I claim as my invention and desire to secure by Letters Patent is:

1. In an apparatus for effecting the transformations of spherical coordinates, a circular base, a member in the shape of a half circle, two bearings carried by the said base, two trunnions carried by the member in the shape of a halfcircle and fitting the respective bearings, the geometrical axis of the bores of the bearings forming a diameter of the base, the geometrical aXis of the trunnions forming a diameter of the member in the shape of a half-circle, a central pivot rigid with the base, asupport rotatably mounted on this pivot, a plate in the shape of a half-circle, carried by this support, the plane of this plate passing through'the axis of the pivot, two cylindrical bored bearings in the support, the geometrical axis of these two bores being at right angles to the plate, trunnions mounted in these bearings, a member in the shape of a half-circle capable of rotating on these trunnions, a rider capable of moving along the outer periphery of this second member in the shape of a half-circle, a rider capable of moving along the inner periphery of the first member in the shape of a half-circle, and a joint connecting the two riders, the geometrical axis of this joint constituting a radius of the two members in the shape of a halfcircle.

2. In an apparatus for efi ecting the trans formations of spherical coordinates, the combination with the means claimed in claim 1, of means for rotating the support on the central pivot of the base, means for rotating the second member in the shape of a half-circle on the trunnions of the support, and means for rotating the first member in the shape of a half-circle in the bearings of the support.

3. In an apparatus for effecting transformations of coordinated spherical bodies, a circular base comprising an annular flange, a member of semi-circular form, two bearings carried by the base, two trunnions carried by the semi-circular member, the geometric axis of the bores of the bearings forming a diameter of the base, the geometric axis of the trunnions forming a diameter of the semi-circular member, a cen tral pivot integral with the socket, a pivotally mounted rotatable support, an arm carried by said support and constructed so that its end moves above the annular flange of the base when the support rotates, a shaft carried by the arm, a gear crown integral with the support, a gear wheel mounted on the shaft, means for rotating said gear wheel, a semi-circular plate carried by the support, the plane of said plate passing through the axis of the pivot, two cylindrical bearings bored into the support, the geo metric axis of these two bearings being perpendicular with the plate and passing through its center, shafts mounted in said bearings, a second semi-circular member mounted on said trunnions and rotatable thereon, a gear crown mounted on the plate, a shaft mounted on the second semi-circular member, a gear wheel mounted on the shaft on the second semi-circular member, means for rotating said last mentioned wheel, a rider movable along the external periphery of the second semi-circular plate, a rider movable along the internal periphery of the first semi-circular member, a joint connecting the two riders, the geometric axis of this pivot constituting a radius of the two semicircular members, a shaft on the first mentioned rider, a gear wheel mounted on the lastmentioned shaft and means for rotating the last mentioned gear wheel.

In testimony whereof I have signed my name to this specification.

CHAR-LES R0 GER MONNEY.

1 is l 

